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Fractional Viscoelastic Models Li Yan Institute of Applied Math, School of Mathematics and System Sciences, Shandong University, 250100, P.R.China. E-mail: yan@cc.usu.edu

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Tao Zhan President & Professor Analysis & Number Theory

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MingYu Xu Professor Theory and Applications of the Fractional Calculus, Biofluid Mechanics, Mathematical Modeling in BME, xumingyu@sdu.edu.cn

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Project Name 国家建设高水平大学公派研究生项目 The project of sending first-class graduate students to study abroad in first-class universities for building high level universities 5000 people’s project

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▪ The Ministry of Education of the P.R.China has launched a five-year (2007-2011) graduate program to send about 5,000 students a year to study in the world's best universities. The most large scale than ever before. ▪ Students will be chosen from the best undergraduates at 49 top universities across the country. ▪ This program is run by China Scholarship Council (CSC).

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Key Research Subjects Energy and natural resources, Environment, Agriculture, Manufacturing, Information technology, Biology, New materials.

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Scholarship Coverage International airfare Health insurance Stipend ($1000-1100/month in USA)

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For more details, please see http://www.eol.cn/liu_xue_kuai_xun_3291/20070404/t2007 0404_226362.shtml

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Fractional Viscoelastic Models Li Yan Institute of Applied Math, School of Mathematics and System Sciences, Shandong University, 250100, P.R.China. E-mail: yan@cc.usu.edu

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Section 1. Introduction ⑴ Riemann-Liouville (R-L) Fractional Operator ⅰ Integral For p>0 In special ⅱ Derivative For p>0

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ⅲ Some Properties of R-L Fractional Operator For p,q>0

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For p,q ∈ R In special where c is a constant

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(2) Generalized Mittag-Leffler Function When β=1 What’s more

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ⅲ Phenomena of Viscoelastic Materials 1. Relaxation and Creep If the strain is held constant, the stress decreases with time (Relaxation) Relaxation Modulus is the stress response of the unit step strain. If the stress is held constant, the strain increases with time (Creep) Creep Modulus is the strain response of the unit step stress. The unit step function is

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2. Hysteresis and Precondition If cyclic loading is applied, Hysteresis (a phase lag) occurs, leading to a dissipation of mechanical energy.

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In strain-stress coordinates

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If cyclic loading is applied, the output will experience a process of “Precondition” and then enter a relative stable condition. 3. Summary of Viscoelastic Phenomena Relaxation, Creep, Hysteresis, Precondition.

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(3) Viscoelastic Models —————Spring-Dashpot models

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ⅰ Spring (Hooke’s Law) where E is the Young’s Modulus. ⅱ Fractional Dashpot When α =1 where η is the viscosity.

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ⅳ Fractional Viscoelastic Models 1. Fractional Maxwell Model Constitutive Equation Relaxation Modulus Creep Modulus

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In Laplace Domain Constitutive Equation Relaxation Modulus Creep Modulus

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2. Fractional Kelvin Model Constitutive Equation Relaxation Modulus Creep Modulus

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In Laplace Domain Constitutive Equation Relaxation Modulus Creep Modulus

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3. Fractional Standard Linear Solid Model Constitutive Equation Relaxation Modulus Creep Modulus

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In Laplace Domain Constitutive Equation Relaxation Modulus Creep Modulus

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4. Constitutive Relationship of Stress and Strain In Laplace domain where ε (0)= σ (0)=0 Moreover

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Hint

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5. Discussion 5-1 Fractional and Integral Maxwell model 5-2 Fractional and Integral Kelvin model

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5-3 The first order derivative of relaxation and creep modulus at original point.

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The unit step function

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5-4 Declining Speed of Power law and M-L Functions. So declines faster than, where λ is an arbitrary positive number.

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What’s more So, for large t

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6. Conclusion · The fractional calculus extends the application of viscoelastic models. · The fractional models are more closely to reality than integral models. · The results of fractional viscoelastic models fit well with experimental methods.

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7. In the future Description of microcosmic systems Simulation of high-speed strain

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Publications Yan Li and Mingyu Xu. Hysteresis and precondition of viscoelastic solid models, Mechanics of time-dependent materials, Vol.10, No.2, June 2006. Yan Li and Mingyu Xu. Hysteresis loop and energy dissipation of viscoelastic solid models, Mechanics of time-dependent materials, Vol.11, No.1, March 2007.

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Thank You !

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